Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $y = \dfrac{8p^2 + 2p}{3} \div \dfrac{12p^2 + 3p}{-2} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{8p^2 + 2p}{3} \times \dfrac{-2}{12p^2 + 3p} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (8p^2 + 2p) \times -2 } { 3 \times (12p^2 + 3p) } $ $ y = \dfrac {-2 \times 2p(4p + 1)} {3 \times 3p(4p + 1)} $ $ y = \dfrac{-4p(4p + 1)}{9p(4p + 1)} $ We can cancel the $4p + 1$ so long as $4p + 1 \neq 0$ Therefore $p \neq -\dfrac{1}{4}$ $y = \dfrac{-4p \cancel{(4p + 1})}{9p \cancel{(4p + 1)}} = -\dfrac{4p}{9p} = -\dfrac{4}{9} $